Volume 2,  Issue 2, 2001

Article 26

A PICK FUNCTION RELATED TO AN INEQUALITY FOR THE ENTROPY FUNCTION

CHRISTIAN BERG

DEPARTMENT OF MATHEMATICS
UNIVERSITY OF COPENHAGEN
UNIVERSITETSPARKEN 5
DK-2100 COPENHAGEN, DENMARK.
E-Mail: berg@math.ku.dk

Received 6 November, 2000; accepted 6 March, 2001.
Communicated by: F. Hansen


ABSTRACT.    The function $ \psi(z)=2/(1+z)+1/($Log$  (1-z)/2)$, holomorphic in the cut plane $ \Bbb C\setminus[1,\infty[$, is shown to be a Pick function. This leads to an integral representation of the coefficients in the power series expansion $ \psi(z)=\sum\limits_{n=0}^\infty\beta_nz^n$, $ \vert z\vert<1$. The representation shows that $ (\beta_n)$ decreases to zero as conjectured by F. Topsøe. Furthermore, $ (\beta_n)$ is completely monotone.
Key words:
Pick functions, completely monotone sequences.

2000 Mathematics Subject Classification:
30E20, 44A60.


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